On the other hand, observations and studies of the two closest black hole low-mass X-ray binaries (BH-LMXBs), A0620-00 and XTE J1118+480, have provided very precise measurements for many important physical parameters, including the orbital period \(P\) , observed radial velocity of the companion star \(K\) , orbital inclination \(i\) , black hole mass \(M_{\rm BH}\) , and the mass of the companion star \(m\)  (or the mass ratio \(q=m/M_{\rm BH}\) ) [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]} (see Table 1 for the measured values). The companion star is orbiting the black hole in a nearly circular orbit for each binary. In particular, observations have revealed abnormally fast orbital decays in the two BH-LMXBs: \(\dot{P}=-0.60 \pm 0.08\)  ms yr\(^{-1}\)  for A0620-00 and \(\dot{P}=-1.90 \pm 0.57\)  ms yr\(^{-1}\)  for XTE J1118+480 [8]}. These decays are two orders of magnitude larger than the one expected with gravitational wave radiation [9]}, [10]}. Standard theories only predict \(\dot{P} \sim -0.02\)  ms yr\(^{-1}\)  [11]}. Two major proposals have been suggested recently to account for the fast orbital decay. The first one is related to the magnetic braking of the companion star. If the surface magnetic field of the companion star is very strong (e.g. \(\ge 10^4\)  G), the coupling between the magnetic field and the winds from the companion star driven by X-ray irradiation from the black hole would decrease the orbital period through tidal torques [9]}, [13]}. However, this model requires a significant mass loss from the binary system, which has not been observed [8]}. The second proposal suggests that the tidal torque between the circumbinary disk and the binary can efficiently extract the orbital angular momentum from the binary to cause the orbital decay [9]}. Nevertheless, simulations show that the predicted mass transfer rate and the circumbinary disk mass are much greater than the inferred values from observations [9]}. Although a few recent studies suggesting the resonant interaction between the binary and a surrounding circumbinary disk could produce the observed orbital period decays [10]}, [18]}, the calculated initial mass and effective temperature of the companion stars somewhat do not match the observations [10]}. Therefore, it is still a mystery for the abnormally fast orbital decays in the two BH-LMXBs.
